Priestly Blessing by a Bat Kohen
(כב) וַיְדַבֵּ֥ר יְהוָ֖ה אֶל־מֹשֶׁ֥ה לֵּאמֹֽר׃ (כג) דַּבֵּ֤ר אֶֽל־אַהֲרֹן֙ וְאֶל־בָּנָ֣יו לֵאמֹ֔ר כֹּ֥ה תְבָרֲכ֖וּ אֶת־בְּנֵ֣י יִשְׂרָאֵ֑ל אָמ֖וֹר לָהֶֽם׃ (ס) (כד) יְבָרֶכְךָ֥ יְהוָ֖ה וְיִשְׁמְרֶֽךָ׃ (ס) (כה) יָאֵ֨ר יְהוָ֧ה ׀ פָּנָ֛יו אֵלֶ֖יךָ וִֽיחֻנֶּֽךָּ׃ (ס) (כו) יִשָּׂ֨א יְהוָ֤ה ׀ פָּנָיו֙ אֵלֶ֔יךָ וְיָשֵׂ֥ם לְךָ֖ שָׁלֽוֹם׃ (ס) (כז) וְשָׂמ֥וּ אֶת־שְׁמִ֖י עַל־בְּנֵ֣י יִשְׂרָאֵ֑ל וַאֲנִ֖י אֲבָרֲכֵֽם׃ (פ)
(22) The LORD spoke to Moses: (23) Speak to Aaron and his sons: Thus shall you bless the people of Israel. Say to them: (24) The LORD bless you and protect you! (25) The LORD deal kindly and graciously with you! (26) The LORD bestow His favor upon you and grant you peace! (27) Thus they shall link My name with the people of Israel, and I will bless them.

(ב) באו ועמדו על מעלות האולם. עמדו הראשונים לדרום אחיהם הכהנים. וחמשה כלים בידם. הטני ביד אחד. והכוז ביד אחד. והמחתה ביד אחד. והבזך ביד אחד. וכף וכסויה ביד אחד. וברכו את העם ברכה אחת. אלא שבמדינה אומרים אותה שלש ברכות ובמקדש ברכה אחת. במקדש היו אומרים את השם ככתבו. ובמדינה בכנויו. במדינה הכהנים נושאים את כפיהם ידיהם כנגד כתפותיהם. ובמקדש על גבי ראשיהן. חוץ מכהן גדול. שאין מגביה את ידיו למעלה מן הציץ. רבי יהודה אומר. אף כהן גדול מגביה את ידיו למעלה מן הציץ. שנאמר (ויקרא ט, כב) וישא אהרן את ידיו אל העם ויברכם:

(2) They [the three priests who did the incense service] would come and stand on the steps of the vestibule. The first [the two who removed the ashes] stood to the south of their colleagues the priests [who did the incense service] and had five utensils in their hands, the basket in the hand of one of them, the jug in the hand of one of them, the firepan in the hand of one of them, the censer in the hand of one of them, and the spoon and its cover in the hand of the other one. And they [all the priests] blessed the nation with one blessing, which in the [outer] districts [outside the temple] are said as three blessings, but in the temple as one blessing. In the temple they would say the [G-d's] name as it is written but outside the temple they would use a pseudonym. Outside the temple the priests lifted their hands [when reciting the blessing] up to their shoulders, but in the temple [they would lift them] above their heads except for the Kohen Gadol who would not lift his hands above the forehead plate [worn by the Kohen Gadol]. Rabbi Yehuda says, even the Kohen Gadol lifts his hands above the forehead plate, as it says, "And Aaron raised his hands towards the people and blessed them."(Leviticus 9:22)

(לט) לֹא הָיוּ בּוֹ מֵהַדְּבָרִים הַמּוֹנְעִים נְשִׂיאַת כַּפַּיִם, אַף עַל פִּי שֶׁאֵינוֹ מְדַקְדֵּק בְּמִצְווֹת וְכָל הָעָם מְרַנְּנִים אַחֲרָיו, נוֹשֵׂא אֶת כַּפָּיו שֶׁאֵין שְׁאָר עֲבֵרוֹת מוֹנְעִין נְשִׂיאַת כַּפַּיִם.

(39) If he does not have any of the disqualifying factors for the priestly blessing, even if he is not careful with mitzvot and the entire congregation complains about him, he may perform the priestly blessing. Ram"a: Because other sins do not inhibit the priestly blessing.

(ב) כָּל כֹּהֵן שֶׁאֵין בּוֹ אֶחָד מֵהַדְּבָרִים הַמְעַכְּבִים, אִם אֵינוֹ עוֹלֶה לַדּוּכָן אַף עַל פִּי שֶׁבִּטֵּל מִצְוַת עֲשֵׂה אַחַת, הֲרֵי זֶה כְּעוֹבֵר בְּג' עֲשֵׂה אִם הָיָה בְּבֵית הַכְּנֶסֶת כְּשֶׁקָּרְאוּ כֹּהֲנִים, אוֹ אִם אָמְרוּ לוֹ לַעֲלוֹת אוֹ לִטֹּל יָדָיו.

(2) Any kohen who does not have one of the inhibiting factors — if he does not ascend to the platform, even though he has [only] forfeited one positive commandment, it is as if he has violated three positive commandments if he was in the synagogue when they called "Kohanim" or if they told him to go up or to wash his hands.

Babylonian Talmud, Tractate Chullin 132b-133a
Rabbi Simeon says: A priest who does not believe in the [Temple] service has no portion in the priesthood, for it is written: He among the sons of Aaron, that offers the blood of the peace-offerings, and the fat, shall have the right thigh for a portion. I only know it with regard to this service; whence do I know it with regard to the fifteen other services in the Temple, viz., the rites of pouring, mingling, breaking into pieces, seasoning with salt, waving, bringing near, taking out the handful, and burning it, the rite of nipping off, of receiving [the blood in a vessel], and sprinkling it, the ceremony of giving the water to a woman suspected of adultery, of breaking the heifer's neck, of purifying the leper, and of raising the hands [for the priestly benediction] both inside (the Temple] and outside? The text therefore states, among the sons of Aaron, that is, every service ordained for the sons of Aaron. Hence, a priest who does not believe in the [Temple] services has no portion in the priesthood. Now the reason for this is that he does not believe in them, but if he does believe in them although he is not conversant with them [he is entitled to the priestly dues].
Teshuvah of Rabbi Joel Roth regarding Bat Kohen and Bat Levi for Aliyot
The Rishonim disagree on the question of whether the daughters of priests may redeem first horns. The Tosafot say: This implies that [Rav Kahana] was not a priest. As regards the report in Kiddushin (8a) that Rav Kahana received payment at a redemption of the first born. That was on behalf of his wife, as it says: Rav Kahana ate on account of his wife.
From this it follows that the daughter of a priest may redeem, and even that her husband may serve as her agent.
The Rosh quotes the same view in the name of Rashi.
On the other hand, Maimonides codifies the law thus: Thus redemption of the first born is for male kohanim, for it says; (Num. 3:51) "You shall give the money to Aaron and his sons."
The same view is expressed by the Rosh himself, and by the Rashba.
Surely according to the Tosafot and the Rosh in Rashi's name, pidyon haben is another indication of lineal sanctity for the daughters of priests, even if married. It is equally important to note, too, that Maimonides' view does not either preclude or deny lineal sanctity to daughters of priests. Their exclusion from the right to redeem is based on the biblical statement, "To Aaron and his sons."

(ו) כהן שלא היה לו דבר מכל אלו הדברים המונעין נשיאת כפים אף ע"פ שאינו חכם ואינו מדקדק במצות או שהיו הבריות מרננים אחריו או שלא היה משאו ומתנו בצדק הרי זה נושא את כפיו ואין מונעין אותו לפי שזו מצות עשה על כל כהן וכהן שראוי לנשיאת כפים ואין אומרים לאדם רשע הוסף רשע והמנע מן המצות.

A priest who does not have any of the things that prevent him from raising his hands, even if he is not wise, and is not observant of the commandments, or people used to speak ill of him, or he was not honest in business-behold he can raise his hands and we do not prevent him because it is a positive commandment for each and every priest that is eligible to raise his hands. We do not say to a wicked person, keep doing evil, and thereby prevent him from doing mitzvot.

(כג) כה תברכו את בני ישראל - כלומר, לא תברכו מברכת פיכם כאדם שאומר תבואתה לראש פלוני כך וכך, אלא אלי תתפללו שאברכם אני, כמו שמפרש - יברכך ה' - ואני אשמע קולכם כאשר תאמרו, ואברכם - לישראל, כמו שמפרש - ושמו את שמי על בני ישראל - כשיברכו הכהנים לישראל בשמי ולא בשמם, אני אברכם לישראל, כמו שהתפללו הכהנים ואומרים: יברכך ה'.
(23) כה תברכו את בני ישראל, you shall not bless them in the manner people bless one another, wishing that certain individuals will experience blessings of a certain kind. Instead, the priests are to make it plain that they pray to G’d to extend His blessings to the people whom the priests face at the time. Verse 24 makes clear that as a result of this prayer to G’d by the priests on behalf of those whom they wish to bless, G’d does in fact extend this blessing. Verse 27 also makes clear that what the priests are instrumental in doing is to bestow G’d’s name on them, not their own. G’d goes on record saying that He would accept the priests’ prayer.

(יג) מִי שֶׁבָּא בַּזְּמַן הַזֶּה וְאָמַר כֹּהֵן אֲנִי אֵינוֹ נֶאֱמָן וְאֵין מַעֲלִין אוֹתוֹ לִכְהֻנָּה עַל פִּי עַצְמוֹ וְלֹא יִקְרָא בַּתּוֹרָה רִאשׁוֹן וְלֹא יִשָּׂא אֶת כַּפָּיו וְלֹא יֹאכַל בְּקָדְשֵׁי הַגְּבוּל עַד שֶׁיִּהְיֶה לוֹ עֵד אֶחָד. אֲבָל אוֹסֵר עַצְמוֹ בִּגְרוּשָׁה וְזוֹנָה וַחֲלָלָה וְאֵינוֹ מִטַּמֵּא לְמֵתִים. וְאִם נָשָׂא אוֹ נִטְמָא לוֹקֶה. וְהַנִּבְעֶלֶת לוֹ סְפֵק חֲלָלָה:

A person who comes to us in our day and age and says, "I am a Cohen." He is not believed, and we do not consider him a Cohen on his own word. And we do not let him read from the Torah first, nor raise his hands, or eat sanctified food-until he has one witness.